# The "symmetric" property of Day convolution.

This question has to be divided into the following parts:

1. The definition of Day convolution in nlab

To define Day convolution, it assumes that $$V$$ be a closed symmetric monoidal category with all small limits and colimits, and $$C$$ be a monoidal category.

Notice that nlab doesn't says that $$C$$ must be symmetric.

2. Day convolution form a monoidal category in nlab

That means, if we there is a tensor unit $$y(I)$$, then the category $$([C,V], ⊗_{Day}, y(I))$$ form a monoidal category automatically.

Notice that nlab doesn't say that $$C$$ must be symmetric.

3. The definition of Day convolution in wikipedia

To define Day convolution, it assumes that $$C$$ be a symmetric monoidal category. (Of course, $$V$$ must be monoidal category, because enriched)

Notice that wikipedia doesn't say that $$V$$ must be symmetric.

4. Day convolution form a monoidal category in wikipedia

It says that

If the category $$V$$ is a symmetric monoidal closed category, we can show this defines an associative monoidal product.

Since a monoidal category must satisfy associative law, that means if we expect that the category $$([C,V], ⊗_{Day}, y(I))$$ form a monoidal category, then $$V$$ must be symmetric, i.e. $$C$$ and $$V$$ are both symmetric monoidal category.

It also provides a proof for this associative law, in which, it seems that the two symmetric /commutative laws be used.

My questions are:

1. Why the definition of Day convolution in nlab and wikipedia are different?

I mean that, to define Day convolution, why nlab require $$V$$ to be a symmetric monoidal category, but wikipedia doesn't require symmetric on $$V$$ and vice versa...

2. Why the condition of "Day convolution form a monoidal category" in nlab and wikipedia are different?

I mean that, to form a monoidal category under Day convolution, why wikipedia require both $$C$$ and $$V$$ are symmetric, but nlab doesn't require this condition?

3. Why Day convolution need some sort of "symmetric" property?

I didn't see any symmetry intuition from this Day convolution formula:

$$F*G = \int^{x,y \in C} C(x \otimes y, -) \otimes Fx \otimes Gy$$

PS: I apologize if the question is silly, I'm a category theory beginner, but this definition make me confusion...

Very thanks.

The description on the nLab is correct: $$\mathscr C$$ does not need to be symmetric, but $$\mathscr V$$ does. If $$\mathscr C$$ is symmetric, then the Day convolution tensor product on $$[\mathscr C, \mathscr V]$$ will also be symmetric. Wikipedia actually does require $$\mathscr V$$ to be symmetric, but delays stating this to establish why symmetry is important: it's necessary for the induced tensor product to be associative (and hence be monoidal). This matches Day's original setting.
As of the time of writing, Wikipedia does state that $$\mathscr C$$ should be symmetric, but this is unnecessary. Anyone can edit Wikipedia, so this could easily be addressed.
• Thanks. I found the wikipedia has been modified. Now to define Day convolution, we just require $(C, \otimes_c)$ be a monoidal category and $(V, \otimes)$ be a symmetric monoidal closed category, right? But the proof of monoidal associative law in wikipedia still need $\otimes_c$ symmetric? see 5th isomorphism, it seems need $\otimes_c$ symmetric? But in nlab, this associative law should not require $\otimes_c$ symmetric. Jun 21 '20 at 15:25
• I have read the proof in this answer, it's great, thanks! So the proof of the associative law doesn't need $\otimes_C$ be symmetric, it just need ninja Yoneda lemma, associativity of $\otimes_C$ and symmetric of $\otimes_V$. So the answer of 3rd question shoube be that the symmetric of $\otimes_V$ can make a monoidal category. Jun 21 '20 at 16:24